Improved Approximation of Phase-Space Densities on Triangulated Domains Using Discrete Flow Mapping with p-Refinement

Abstract

We consider the approximation of the phase-space flow of a dynamical system on a triangulated surface using an approach known as Discrete Flow Mapping. Such flows are of interest throughout statistical mechanics, but the focus here is on flows arising from ray tracing approximations of linear wave equations. An orthogonal polynomial basis approximation of the phase-space density is applied in both the position and direction coordinates, in contrast with previous studies where piecewise constant functions have typically been applied for the spatial approximation. In order to improve the tractability of an orthogonal polynomial approximation in both phase-space coordinates, we propose a careful strategy for computing the propagation operator. For the favourable case of a Legendre polynomial basis we show that the integrals in the definition of the propagation operator may be evaluated analytically with respect to position and via a spectrally convergent quadrature rule for the direction coordinate. A generally applicable spectral quadrature scheme for integration with respect to both coordinates is also detailed for completeness. Finally, we provide numerical results that motivate the use of p-refinement in the orthogonal polynomial basis.

Wave scattering properties of multiple weakly coupled complex systems

“…In the Power Balance Method the mean power flow through connected over-moded cavities is calculated [22,33,34]. The Dynamic Energy Analysis (DEA) method involves solving for the phase-space energy density on a gridded domain [35], and describes mean high-frequency wave energy distributions in all sub-systems [36,37]. The Random Coupling Model (RCM) determines the statistical properties of the impedance and scattering parameters for complex enclosures [38][39][40][41][42][43][44].…”

High-frequency structure- and air-borne sound transmission for a tractor model using Dynamical Energy Analysis

“…where 1 b is the characteristic function of edge b (that is, 1 b (s) = 1 for s on edge b and 0 elsewhere) and A b is the length of edge b. For an implementation using higher order basis functions in position space, see [12]. The multiindex n ≡ (b, β) combines the index of the edge b with the index β, which corresponds to the degree of the Legendre polynomial.…”

“…Figure A.14: Transmission and reflection coefficients for two plates at a 90 • angle. The different coefficients are added up, so |S12 BL | 2 is the distance from the red curve to the blue one. α is the angle of the incoming ray.…”

Efficient Statistical Model for Predicting Electromagnetic Wave Distribution in Coupled Enclosures

“…The PWB method is based on the assumption of a uniform field distribution in each cavity which is often fulfilled in the weak inter-cavity coupling and low damping limit. Extensions of the PWB method that drop the uniform field assumption are ray tracing (RT) methods [35,36] or the Dynamical Energy Analysis (DEA) method [37][38][39] which calculate local ray or energy densities. DEA and RT capture non-uniformity in the field distribution within a given sub-volume.…”